In parallelogram ABCD, the diagonals intersect at point E, point M is taken in the middle

In parallelogram ABCD, the diagonals intersect at point E, point M is taken in the middle of side AB. Segment DM intersects the diagonal AC at point O. Find AO and OC if AC = 18 cm.

For triangle ABD, the segment AE is the median, because it divides the side of BD at point E in half (the diagonals intersect in the middle).

Point M is in the middle of side AB, and therefore the segment MD is also the median of triangle ABD. The intersection medians are divided in a ratio of 2: 1.

The segment AE is equal to half of the diagonal AC:

AE = AC / 2;

AO / OE = 2/1;

AO = 2 * OE;

AE = AO + OE = 2 * OE + OE = 3 * OE;

OE = AE / 3 = (AC / 2) / 3 = AC / 6 = 18/6 = 3 cm;

AO = 3 * 2 = 6 cm.

Answer: AO = 6 cm, OE = 3 cm.